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| Mirrors > Home > MPE Home > Th. List > reupick3 | Structured version Visualization version GIF version | ||
| Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.) |
| Ref | Expression |
|---|---|
| reupick3 | ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 → 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-reu 3381 | . . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 2 | df-rex 3071 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))) | |
| 3 | anass 468 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))) | |
| 4 | 3 | exbii 1848 | . . . . 5 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))) |
| 5 | 2, 4 | bitr4i 278 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓)) |
| 6 | eupick 2633 | . . . 4 ⊢ ((∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)) | |
| 7 | 1, 5, 6 | syl2anb 598 | . . 3 ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)) |
| 8 | 7 | expd 415 | . 2 ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) → (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))) |
| 9 | 8 | 3impia 1118 | 1 ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 → 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 ∃wex 1779 ∈ wcel 2108 ∃!weu 2568 ∃wrex 3070 ∃!wreu 3378 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-ex 1780 df-mo 2540 df-eu 2569 df-rex 3071 df-reu 3381 |
| This theorem is referenced by: reupick2 4331 fvineqsneq 37413 |
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